(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 24857, 793]*) (*NotebookOutlinePosition[ 25523, 816]*) (* CellTagsIndexPosition[ 25479, 812]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ StyleBox[\(Mathematica\ Tutorial\), FontFamily->"Courier New", FontSize->26, FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "StrikeThrough"->False}]], "Input"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Welcome to ", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica ", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox["!!! To evaluate the first 4 cells below, place your cursor at \ the end of each expression, hold down the ", FontColor->RGBColor[0, 0, 1]], StyleBox["Shift", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" key, and hit ", FontColor->RGBColor[0, 0, 1]], StyleBox["Enter.", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" Once you have completed these, try a few expressions of your \ own.", FontColor->RGBColor[0, 0, 1]] }], "Section"], Cell[BoxData[ \(5 + 6\)], "Input"], Cell[BoxData[ \(8*9\)], "Input"], Cell[BoxData[ \(4 - 5*6 + 5/9\)], "Input"], Cell[BoxData[ \(\((87\ - \ 45)\)/\((3*\((\(-4\) + 17\ *\ 13)\)\ + \ 82)\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Next, evaluate the following expressions. Again, move the cursor \ to the end of the expression, hold down the ", FontColor->RGBColor[0, 0, 1]], StyleBox["Shift", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" key, and hit ", FontColor->RGBColor[0, 0, 1]], StyleBox["Enter. Note that % means \"the previous result\".", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]] }], "Section"], Cell[BoxData[ \(\@44\)], "Input"], Cell[BoxData[ \(3\^7\)], "Input"], Cell[BoxData[ \(\((4 + 5 \[ImaginaryI])\)\^2\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"(", GridBox[{ {"3", \(-4\)}, {\(\(\ \)\(6\)\), "9"} }], ")"}], ".", RowBox[{"(", GridBox[{ {\(-1\), "2"}, {"5", \(-7\)} }], ")"}]}], Cell[""]}]], "Input"], Cell[BoxData[ RowBox[{\({{3, \(-4\)}, {6, 9}}\), " ", "\[Equal]", " ", RowBox[{"(", GridBox[{ {"3", \(-4\)}, {\(\(\ \)\(6\)\), "9"} }], ")"}]}]], "Input"], Cell[BoxData[ \(\[Sum]\+\(k\ = \ 1\)\%40 k\^2\)], "Input"], Cell[BoxData[ \(Sum[k^2, {k, 1, 40}]\ \[Equal] \ %\)], "Input"], Cell[BoxData[ \(\[Integral]\(4\/\(3\ + \ 4 x\^2\)\) \[DifferentialD]x\)], "Input"], Cell[BoxData[ \(\[Integral]\+2\%3 Sin[2 x] \[DifferentialD]x\)], "Input"], Cell[BoxData[ \(Sec[\(5 \[Pi]\)\/3]\ + \ Cot[\(7 \[Pi]\)\/6]\)], "Input"], Cell[BoxData[ \(\[Sum]\+\(i\ = \ 2\)\%47 1\/\(i + \ 3\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Notice that when you are working with integers,", StyleBox["Mathematica", FontSlant->"Italic"], " provides output with integers as well. Sometimes this might not be the \ output you are looking for. When you want decimal approximations, use the \ built-in function N[ expression], or N[expression,precision]. Try a few of \ the following." }], "Section", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(\@32\)], "Input"], Cell[BoxData[ \(N[\@32]\)], "Input"], Cell[BoxData[ \(\@32 // N\)], "Input"], Cell[BoxData[{ \(Cos[\[Pi]\/6]\ + \ Sec[\(7 \[Pi]\)\/6]\), "\[IndentingNewLine]", \(Cos[\[Pi]\/6]\ + \ Sec[\(7 \[Pi]\)\/6] // N\)}], "Input"], Cell[BoxData[{ \(\[Sum]\+\(k\ = \ 1\)\%52 2\/k\^3\), "\n", \(N[\ \[Sum]\+\(k\ = \ 1\)\%52 2\/k\^3]\), "\n", \(N[\[Sum]\+\(k\ = \ 1\)\%52 2\/k\^3, \ 30]\)}], "Input"], Cell[BoxData[ \(N[\[Pi], 40]\)], "Input"], Cell[BoxData[ \(N[\[ExponentialE]\^2, 26]\)], "Input"] }, Open ]], Cell["\<\ Now, how do we get all of those pretty symbols, like \[Pi], \[ExponentialE], \ \[ImaginaryI], \[Sum], and \[Integral] , among others? One way to get these \ symbols is through the use of palettes. Using your cursor, move to the menu \ \"File\", then \"Palettes\", and then \"BasicInput\". You might also want to \ investigate \"BasicTypesetting\" and \"BasicCalculations\". Try and \ reproduce some of the expressions that you see above, or variations on these \ expressions.\ \>", "Section", FontColor->RGBColor[0, 0, 1]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" can work with expressions as well as numerical input. You can \ factor, combine like terms, and expand expressions. Try the following, then \ try a few on your own.", FontColor->RGBColor[0, 0, 1]] }], "Section"], Cell[BoxData[ \(4 x\^2\ - \ 3 x\ + \ 7\ - \ 8 x\^2\ + \ 6 x\^3\ + \ 11 x\ - \ 9\)], "Input"], Cell[BoxData[{ \(Simplify[\((7 x\^2 + \ 3 x\ - \ 8)\)\ - \ \((6 x\^2\ - \ 5 x\ - \ 24)\)]\), "\[IndentingNewLine]", \(\((7 x\^2 + \ 3 x\ - \ 8)\)\ - \ \((6 x\^2\ - \ 5 x\ - \ 24)\) // Simplify\)}], "Input"], Cell[BoxData[{ \(Expand[\((2 x\^2 - 3 x + \ 4)\)\^4]\), "\[IndentingNewLine]", \(\((2 x\^2 - 3 x + \ 4)\)\^4 // Expand\)}], "Input"], Cell[BoxData[ \(2 x\^3 + \ 13 x\^2\ - \ 7 x // Factor\)], "Input"], Cell[BoxData[ \(\(5 x + 2\)\/\(x\^2 + \ 5 x\ + \ 4\) // Apart\)], "Input"], Cell[BoxData[ \(1\/\(2 x + 3\) - \ \(5 x\)\/\(x\^2 - 1\) // Together\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["A nice utility in ", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" is the % expression, which represents the last output. Test it \ by evaluating the expressions below, in order (don't skip any).", FontColor->RGBColor[0, 0, 1]] }], "Section"], Cell[BoxData[ \(\(4 x\^2 - 5\)\/\(2 x\^2 - 5 x + 2\)\)], "Input"], Cell[BoxData[ \(Apart[%]\)], "Input"], Cell[BoxData[ \(Together[%]\)], "Input"], Cell[BoxData[ \(Expand[%]\)], "Input"], Cell[BoxData[ \(ExpandAll[%]\)], "Input"], Cell[BoxData[ \(Simplify[%]\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{ StyleBox["A", FontFamily->"Courier New", FontSize->24, FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "StrikeThrough"->False}], StyleBox[" ", FontFamily->"Courier New", FontSize->24, FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "StrikeThrough"->False}], StyleBox["list", "Subtitle", FontFamily->"Courier New", FontSize->24, FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, 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FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "StrikeThrough"->False}]}], StyleBox[",", FontFamily->"Courier New", FontSize->24, FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "StrikeThrough"->False}], StyleBox[" ", FontFamily->"Courier New", FontSize->24, FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "StrikeThrough"->False}], StyleBox[\(enclosed\ in\ curly\ brackets . \ Most\ Mathematica\ operations\ operate\ on\ lists\ as\ well\ as\ \ single\ \(\(expressions\)\(.\)\)\), FontFamily->"Courier New", FontSize->24, FontWeight->"Bold", FontSlant->"Plain", FontColor->RGBColor[0, 0, 1], FontVariations->{"Underline"->False, "StrikeThrough"->False}]}]], "Input"], Cell[BoxData[ \(\(L\ = \ {1, 2, 3, 4, 5};\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(L/2\)], "Input"], Cell[BoxData[ \({1\/2, 1, 3\/2, 2, 5\/2}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(2/L\)], "Input"], Cell[BoxData[ \({2, 1, 2\/3, 1\/2, 2\/5}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(L\^2\)], "Input"], Cell[BoxData[ \({1, 4, 9, 16, 25}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Log[L]\^3 // N\)], "Input"], Cell[BoxData[ \({0.`, 0.33302465198892944`, 1.3259689601439077`, 2.6641972159114355`, 4.168911564285652`}\)], "Output"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" can \"hold\" values in variables, and this works as you might \ expect. Evaluate the Cells below in order. Note that if there is a \ semicolon after a statement, the output is suppressed (but the variable value \ is still assigned). Also note that the variables are \"cleared\" at the end \ so that they are ready to be reassigned later.", FontColor->RGBColor[0, 0, 1]] }], "Section"], Cell[BoxData[ \(\(m\ = \ 5;\)\)], "Input"], Cell[BoxData[ \(n\ = \ 6; \ t\ = \ 9.4; \ s\ = \ \(-3.345677\);\)], "Input"], Cell[BoxData[ \(m\ *\ n\)], "Input"], Cell[BoxData[ \(m\ - \ n\ *\ t\ /\ s\^2\)], "Input"], Cell[BoxData[ \(t\)], "Input"], Cell[BoxData[ \(Clear[m, n, t, s]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ The syntax for defining functions is a little less intuitive, and you will \ want to take note of the x_ (that's x with an underscore after it), the \ bracket notation [ ] (not parantheses), and the := (not just an equals sign, \ but colon equals). Evaluate the following expressions, and then try some of \ your own.\ \>", "Section", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(f[x_]\ := \ x\^2 - 5 x + 7\)], "Input"], Cell[BoxData[ \(g[x_] := \ Sin[2 x\ - \ \[Pi]\/3]\)], "Input"], Cell[BoxData[ \(f[1]\)], "Input"], Cell[BoxData[ \(f[5] + g[\[Pi]\/3]\)], "Input"], Cell[BoxData[ \(g[a\^2 - a]\)], "Input"], Cell[BoxData[ \(f[5 - 2 \[ImaginaryI]]\)], "Input"], Cell[BoxData[ \(f[g[\[Pi]\/6]]\)], "Input"], Cell[BoxData[ \(f[g[x]]\)], "Input"], Cell[BoxData[ \(Clear[f, g]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "Setting up expressions and then substituting also has its quirks. See the \ expression below, expr /. y \[Rule] 2, which you should read as \"replace y \ in expr with 2\". Be careful, too, with \"implied multiplication\". You \ need to make sure that you place spaces between the variables for ", StyleBox["Mathematica", FontSlant->"Italic"], " to understand this. Evaluate and then try your own." }], "Section", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(expr\ = \ x\ y\ z\ + \ x\^2 + \ 3 y\^2\ + \ 2 y\ z\)], "Input"], Cell[BoxData[ \(expr\ /. y \[Rule] 2\)], "Input"], Cell[BoxData[ \(expr\ /. \ x\ \[Rule] \ \(-3\)\)], "Input"], Cell[BoxData[ \(expr\ /. \ {x\ \[Rule] \ 2, \ z\ \[Rule] \ \(-1\)}\)], "Input"], Cell[BoxData[ \(expr\ /. \ {x\ \[Rule] \ 5, \ y\ \[Rule] \ \(-4\), \ z\ \[Rule] \ 3}\)], "Input"] }, Open ]], Cell[BoxData[ \(Clear[expr]\)], "Input"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " can also solve polynomial equations in one variable up to the 4th degree. \ Use one of the functions Solve or NSolve. Take note of the double equals \ sign (==). You may also add a second argument to these functions to indicate \ what variable you wish to solve for. Try a few of your own after evaluating \ the following." }], "Section", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(Solve[x\^2 - 4 x - 5 \[Equal] 0]\)], "Input"], Cell[BoxData[ \(Solve[x\^2 - 4 x + 5 \[Equal] 0]\)], "Input"], Cell[BoxData[ \(NSolve[\(1\/5\) x\^4 + \(1\/3\) x\^3 + 2 x - 7 \[Equal] 0]\)], "Input"], Cell[BoxData[ \(NSolve[0.1 x\^3 - x\^2 - 3 x + 4 \[Equal] 0]\)], "Input"], Cell[BoxData[ \(Solve[x\ y\ + \ 3 x\^2\ y\ - \ y\^2 \[Equal] 0, \ x]\)], "Input"], Cell[BoxData[ \(NSolve[x\^4 - \@\(2 x\) + 3 x \[Equal] 5, x]\)], "Input"], Cell["\<\ You can verify your results by plugging your solutions back into the \ function.\ \>", "Section", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(f[x_]\ := x\^4 + 6 x\^3 - 11 x\^2 - 24 x + 28\)], "Input"], Cell[BoxData[ \(solutions\ = \ Solve[f[x] \[Equal] 0, x]\)], "Input"], Cell[BoxData[ \(TableForm[solutions]\)], "Input"], Cell[BoxData[ \(f[x] /. solutions\)], "Input"], Cell["\<\ Solving a system of equations requires that you plug in more than one \ equation, and then list the variables you wish to solve for.\ \>", "Section", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(Solve[{2 x + 3 y \[Equal] 6, y - 3 x \[Equal] 5}, {x, y}]\)], "Input"], Cell[BoxData[ \(Solve[{x\^2 + y\^2 \[Equal] 4, y \[Equal] x\^2 - 3}, {x, y}]\)], "Input"], Cell[BoxData[ \(Solve[{x + 3 y + 5 z \[Equal] 9, x - 2 y + 3 z \[Equal] 2, \(-2\) x + 3 y - 4 z \[Equal] \(-3\)}, {x, y, z}]\)], "Input"], Cell[TextData[{ StyleBox["One of the best things about ", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" is its graphics capabilities. Evaluate the following, and then \ replace some of the functions with your own creations. You can also resize \ the output (graph) by clicking on the image, then clicking and dragging on \ the corners.", FontColor->RGBColor[0, 0, 1]] }], "Section"], Cell[BoxData[ \(Plot[\(1\/2\) x\^3 - \ 7 x\ + \ 4, \ {x, \(-5\), 6}]\)], "Input"], Cell[BoxData[ \(Plot[x*Sin[2 x], \ {x, \ \(-8\) \[Pi], \ 2 \[Pi]}]\)], "Input"], Cell[BoxData[ \(Plot[\[ExponentialE]\^\(\(2\/3\) x\), \ {x, \ \(-5\), \ 2}]\)], "Input"], Cell[BoxData[ \(Plot[{Cos[2 x], 2*Sin[x]}, \ {x, \(-4\) \[Pi], \ 4 \[Pi]}]\)], "Input"], Cell[BoxData[ \(Plot[{\(1\/3\) x\^3, \ x\^2, \ 2 x}, \ {x, \(-3\), 3}]\)], "Input"], Cell[TextData[{ StyleBox["For any built-in ", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" function, you can find out more by entering ?function or \ Options[function]. Learn more about the Plot function by evaluating the \ Cells below.", FontColor->RGBColor[0, 0, 1]] }], "Section"], Cell[BoxData[ \(\(?Plot\)\)], "Input"], Cell[BoxData[ \(Options[Plot]\)], "Input"], Cell[TextData[{ StyleBox["Wow!! That's a lot of different options for the Plot function. \ Another (and more informative) way of learning about a built-in function is \ to go to the \"Help\" menu above, choosing \"Help Browser..\", and then \ typing in the name of the function, then hitting ", FontColor->RGBColor[0, 0, 1]], StyleBox["Enter", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[". Try this for the Plot function. You will find examples, \ related functions, a description of options, etc. Let's finish with a few \ more Plot examples, and try to change some of the options so that you can see \ how it affects the plot. 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